Metatron Dev. IV: Path Guiding

ReSTIR PG
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局部路径引导拟合的理想分布为后缀贡献:

\begin{equation} p(\omega_i|\mathbf{p}, \omega_o) \propto f(\mathbf{p}, \omega_o, \omega_i)L_i(\mathbf{p}, \omega_i)|\cos\theta| \end{equation}

ReSTIR PT将路径贡献作为目标分布 $\hat{p}(\mathbf{x})=f(\mathbf{x})$ , 记 $\mathcal{A}$ 为场景表面, $W_e$ 为传感器响应, $h_x$ 为像素 $x$ 的重建滤波器, $L_e(\mathbf{p}_n)$ 为末端顶点自发光, 路径贡献函数为:

\begin{equation} f(\mathbf{x}) = W_e(\mathbf{p}_0 \to \mathbf{p}_1)G(\mathbf{p}_0 \leftrightarrow \mathbf{p}_1) \prod_{i=1}^{n-1} f(\mathbf{p}_{i+1} \to \mathbf{p}_i \to \mathbf{p}_{i-1})G(\mathbf{p}_i \leftrightarrow \mathbf{p}_{i+1}) L_e(\mathbf{p}_n) \end{equation}

定义逐顶点的传输因子 $T$ 并简化上式:

\begin{equation} T(\mathbf{p}_i) = \begin{cases} f(\mathbf{p}_{i+1} \to \mathbf{p}_i \to \mathbf{p}_{i-1})G(\mathbf{p}_i \leftrightarrow \mathbf{p}_{i+1}), & i > 1\\ W_e(\mathbf{p}_0 \to \mathbf{p}_1)G(\mathbf{p}_0 \leftrightarrow \mathbf{p}_1)f(\mathbf{p}_2 \to \mathbf{p}_1 \to \mathbf{p}_0)G(\mathbf{p}_1 \leftrightarrow \mathbf{p}_2), & i = 1 \end{cases} \end{equation}

\begin{equation} f([\mathbf{p}_0, \dots, \mathbf{p}_n]) = \prod_{i=1}^{n-1}T(\mathbf{p}_i)\ L_e(\mathbf{p}_n) \end{equation}

记像素 $x$ 的路径空间为 $\Omega_x$ :

\begin{equation} C_x = \frac{1}{\int_{\Omega_x} f(\mathbf{x})\mathrm{d}\mathbf{x}} \end{equation}

全体像素构成对整个路径空间 $\Omega=\bigcup_{n=2}^\infty \mathcal{A}^n$ 的分层抽样, 其密度为各像素分布的混合:

\begin{equation} p(\mathbf{x}) = \Phi(\mathbf{p}_0, \mathbf{p}_1)f(\mathbf{x}), \quad \Phi(\mathbf{p}_0, \mathbf{p}_1) = \frac{1}{N}\sum_{x=1}^N C_x[h_x(\mathbf{p}_0 \to \mathbf{p}_1) > 0] \end{equation}

记 $\mathrm{d}\mathbf{p}_{a:b}=\mathrm{d}\mathbf{p}_a \cdots \mathrm{d}\mathbf{p}_b$ , 约定 $a > b$ 时积分退化为被积函数本身:

\begin{equation} p(\mathbf{p}_{i+1}|\mathbf{p}_0, \dots, \mathbf{p}_i) = \frac{p(\mathbf{p}_0, \dots, \mathbf{p}_{i+1})}{p(\mathbf{p}_0, \dots, \mathbf{p}_i)} = \frac{\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i-1}} p(\mathbf{x})\mathrm{d}\mathbf{p}_{i+2:n}} {\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i}} p(\mathbf{x})\mathrm{d}\mathbf{p}_{i+1:n}} \end{equation}

$\Phi$ 只依赖 $\mathbf{p}_0$ 与 $\mathbf{p}_1$ , 条件 $i \geq 1$ 已固定二者, 同时 $\prod_{t=1}^{i-1}T(\mathbf{p}_t)$ 与积分变量无关, 一并约去:

\begin{equation} \begin{aligned} p(\mathbf{p}_{i+1}|\mathbf{p}_0, \dots, \mathbf{p}_i) &= \frac{\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i-1}} \prod_{t=1}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+2:n}} {\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i}} \prod_{t=1}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+1:n}}\\ &= \frac{T(\mathbf{p}_i)\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i-1}} \prod_{t=i+1}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+2:n}} {\sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i}} \prod_{t=i}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+1:n}} \end{aligned} \end{equation}

分子中 $T(\mathbf{p}_i)$ 之后的求和即入射辐亮度:

\begin{equation} L_i(\mathbf{p}_{i+1} \to \mathbf{p}_i) = \sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i-1}} \prod_{t=i+1}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+2:n} \end{equation}

分母相比入射辐亮度少了自发光:

\begin{equation} L_i(\mathbf{p}_i \to \mathbf{p}_{i-1}) - L_e(\mathbf{p}_i \to \mathbf{p}_{i-1}) = \sum_{n=i+1}^\infty \int_{\mathcal{A}^{n-i}} \prod_{t=i}^{n-1}T(\mathbf{p}_t)L_e(\mathbf{p}_n)\mathrm{d}\mathbf{p}_{i+1:n}\\ \end{equation}

分母只依赖 $\mathbf{p}_{i-1}$ 与 $\mathbf{p}_i$ , 相对 $\mathbf{p}_{i+1}$ 为常数, 因此 $\mathbf{p}_0, \dots, \mathbf{p}_{i-2}$ 对条件分布没有影响:

\begin{equation} p(\mathbf{p}_{i+1}|\mathbf{p}_{i-1}, \mathbf{p}_i) \propto f(\mathbf{p}_{i+1} \to \mathbf{p}_i \to \mathbf{p}_{i-1})G(\mathbf{p}_i \leftrightarrow \mathbf{p}_{i+1})L_i(\mathbf{p}_{i+1} \to \mathbf{p}_i) \end{equation}

转立体角测度结果如下, 即服从路径贡献分布的路径, 局部弹射方向的条件分布为理想的局部引导分布. ReSTIR样本只是无偏权重, 初始样本质量差时相关性较强.

\begin{equation} p(\omega_i|\mathbf{p}_i, \omega_o) \propto f(\mathbf{p}_i, \omega_o, \omega_i)L_i(\mathbf{p}_i, \omega_i)|\cos\theta| \end{equation}

$p(\omega_i|\mathbf{p}, \omega_o)$ 需要拟合7D模型因此实时不可行. 划分场景为空间网格, 在格内拟合求平均以消去 $\mathbf{p}$ , 降维到4D的 $p(\omega_i|\omega_o)$ . 漫反射BSDF为常数, 光滑镜面更依赖BSDF抽样, 因此通过积分消去 $\omega_o$ 后, 只有粗糙镜面效果较差, 可进一步降维到2D:

\begin{equation} p(\omega_i) = \int_{\mathcal{H}^2} p(\omega_i|\omega_o)p(\omega_o)\mathrm{d}\omega_o \propto L_i(\omega_i)|\cos\theta| \int_{\mathcal{H}^2} f(\omega_o, \omega_i)p(\omega_o)\mathrm{d}\omega_o \end{equation}

ReSTIR PG #

ReSTIR PG
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